Download Curie-Weiss law in thin-film ferroelectrics

JOURNAL OF APPLIED PHYSICS 100, 044114 共2006兲
Curie-Weiss law in thin-film ferroelectrics
Biao Wang
School of Physics and Engineering, Sun Yat-sen University, Guangzhou, China
C. H. Wooa兲
Department of Electronic and Information Engineering, The Hong Kong Polytechnic University,
Hung Hom, Kowloon, Hong Kong
共Received 20 July 2005; accepted 17 July 2006; published online 31 August 2006兲
The stationary self-polarization field of a thin film in an open circuit is analytically solved for
temperatures near the para-/ferroelectric transformation within the Ginzburg-Landau theory. For
second-order ferroelectrics, or first-order ferroelectrics with a sufficiently large elastic self-energy of
the transformation strain, the solution is real and stable, from which the corresponding electric
susceptibility of the film can be derived. A Curie-Weiss-type relation of the permittivity is obtained
for both the supercritical and subcritical temperature regimes near the transition. In the paraelectric
state, the Curie parameter of the thin film is found to be independent of its thickness, whereas in the
ferroelectric state, its magnitude decreases rapidly with decreasing film thickness. © 2006 American
Institute of Physics. 关DOI: 10.1063/1.2336979兴
I. INTRODUCTION
The large surface to bulk ratio causes ferroelectric thin
films to exhibit phase-transition characteristics that are generally different from their bulk counterparts. Indeed, in extreme cases, such as in PbTiO3, BaTiO3, and lead zirconic
titanate 共PZT兲 thin films, the ferroelectric transition can even
be totally suppressed, when the film thickness is below certain critical values. This dependence of the phase-transition
characteristics on the sample size has been investigated using
both first-principles calculations1,2 and thermodynamic
models.3–5 Experiments have also confirmed that the Curie
temperatures of epitaxial films of PbTiO3 grown on SrTiO3
共001兲 substrate decrease rapidly with the film thickness below 50 nm, as predicted theoretically.5,6
Due to the ever-increasing size and complexity of electronic circuitry that has to be packed into the limited space of
modern integrated circuit 共ICs兲, the influence of sample size
on material properties is a subject of general interest to both
scientific and technological investigations.10–12 For the general phase-transition problem of finite size systems, some
scaling laws have been established,7–9 and it is not unusual
that controlling parameters for phase transitions such as the
Curie temperature may differ substantially from those in the
bulk materials. In addition, important physical properties
may also become sensitive to sample dimensions through the
stability of the prevailing phase.
In a previous paper,5 we studied the relationship between
the film thickness and the Curie temperature of the para-/
ferroelectric transition in a thin film on a rigid substrate. By
considering the dynamic stability of the stationary solution
of the Ginsberg-Landau equation, simple expressions of the
Curie temperature as a function of film thickness were derived without having to go through the complexities of an
explicit solution. The order of the transition and the difference between the heat-up and cool-down transition temperaa兲
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tures, on the other hand, were found to be independent of the
film thickness, but might depend on the constraint of the
substrate. This conclusion was further investigated in another
work,13 in which the Ginzburg-Landau equation for a thin
film on a substrate with varying compliance was solved in
the neighborhood of the transition temperature. Although the
conclusion of Ref. 5 was confirmed, yet for the case of a
compliant substrate, Ref. 13 found that the order of transition
of a thin film of first-order ferroelectric material did not remain constant, but may indeed change according to the film
thickness relative to that of the substrate.
In the present paper, we are interested in the effect of the
film thickness on the Curie-Weiss law of electric susceptibility. In this calculation, the explicit solution of the GinzburgLandau equation is required, which we obtained via a perturbation series expansion method.13 In Sec. II, the evolution of
the ferroelectric thin films is formulated following Wang and
Woo5 within the dynamic Ginzburg-Landau theory using the
corresponding free energies, in which the local selfpolarization near a free surface is subsumed in the boundary
conditions through the “extrapolation length” parameter. The
constraint of a rigid substrate is assumed, and we consider
electric boundary conditions under which the film is between
electrodes from which it is insulated. In Sec. III, the stationary solutions of the Ginzburg-Landau equation are obtained
in the neighborhood of the transition temperature, by means
of a perturbation series expansion method. The solutions are
then used in our investigation in Sec. IV. The paper is summarized and concluded in Sec. V.
II. FORMULATION
We consider a ferroelectric thin film of dimensions ⬁
⫻ ⬁ ⫻ h 共i.e., h = film thickness兲 on a rigid substrate, i.e., one
with a thickness much larger than h. The origin of the coordinate system is at the center of the cell. We base our formulation on the classical description of electrical susceptibility
of a collection of polar molecules, the local polarization Ptotal
100, 044114-1
© 2006 American Institute of Physics
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044114-2
J. Appl. Phys. 100, 044114 共2006兲
B. Wang and C. H. Woo
at any point consists of a permanent molecular component P
and an induced component PE. P arises from the permanent
electric moment due to the polar nature of the molecular
structure/configuration of the material, e.g., its crystallography, in its prevailing para-/ferroelectric phase. P is thus a
function of the temperature, the electric field and the mechanical stresses in the sample through its Curie temperature.
We call P the self-polarization, instead of “spontaneous polarization” like what we used to Ref. 5, to distinguish it from
the more commonly used meaning of the total 共i.e., observed兲 polarization in the absence of an external field. We
note that Ptotal and P are equal only when the depolarization
field vanishes, such as in the bulk material between shortcircuited electrodes.
The induced component PE is the polarizations 共ionic
+ electronic兲 induced by the total electric field E in the material according to PE = ␹dE, where ␹d is the temperatureindependent component of the susceptibility corresponding
to the field-induced part of the polarization. We may identify
E, which has its origin from the combined effects of P and
Ex, as the sum of the external field Ex and the depolarization
field Ed. In this regard, the displacement field in the ferroelectric can be written either in terms of Ptotal or P:
D = ␧0E + Ptotal = ␧0E + P + ␹dE = ␧dE + P,
where ␧0 is the background permittivity, and ␧d is that part of
the permittivity of the ferroelectric corresponding to ␹d.
For simplicity the nonzero component of the selfpolarization P of the ferroelectric thin film is assumed to be
T
orthogonal to the surface of the film. The eigenstrain ␧xx
T
2
= ␧yy = QP due to the para-/ferroelectric transition is constrained by the rigid substrate, causing an elastic self-energy
2
/ C11兲, C11
Fe = GQ2 兰 兰兰V P4d␯, where G = 共C11 + C12 − 2C12
and C12 being components of the elastic modulus, and V the
volume, of the film. We neglect the effects of the epitaxial
stresses to simplify the discussion in the present work. It is
clear from Ref. 5 that its inclusion only has the equivalent
effect of shifting the bulk transition temperature. Following
Ref. 5, with the help of the Ginzburg-Landau functional, the
dynamic equation of the self-polarization in the thin film can
be written as
⳵P
␦F
=−
= − A共T − TC0兲P − 共B + 4GQ2兲P3 − CP5
M
⳵t
␦P
d2 P
+ Ed + D 2 ,
dz
共1兲
with the boundary conditions,
P
dP
= ⫿ ,
dz
␦
h
z= ± ,
2
共2兲
where M is the kinetic coefficient related to the domain wall
mobility; A, B, C, and D, are expansion coefficients of the
Ginzburg-Landau functional of the homogeneous selfpolarization in the bulk ferroelectric; TC0 is the corresponding Curie temperature; Ed is the depolarizing field, and ␦ is
the extrapolation length that describes the near-surface relaxation effect of the local self-polarization.14,15 The meaning of
the other symbols are as given in Ref. 5.
The electric and polarization fields in the ferroelectric
are related through conditions on the boundaries via the laws
of electrostatics. In this paper, a ferroelectric thin film in an
open circuit is considered, i.e., the film is insulated from the
electrodes. In this case, the depolarization field Ed has been
derived5
Ed = −
P
.
␧d
共3兲
Note that the Ed here is different from and generally much
larger than that in Ref. 15.
Most of bulk ferroelectrics exhibit a first-order phase
transition. In such cases, the parameter B is negative, and the
parameter C is positive. For bulk materials in which the
ferroelectric transition is second order, the parameter B is
positive and the higher order terms can be neglected.
III. STEADY-STATE SOLUTION NEAR THE
BIFURCATION POINT
The condition of dynamic stability of the paraelectric
state of the evolution equation Eq. 共1兲 has been established,5
from which the critical temperature Tc at which the state
becomes unstable can be obtained as a function of the film
thickness h:
Tc = TC0 −
1
D
− kz2 min ,
A␧d A
共4兲
where kz min is the minimum solution of the following transcendental equation:
冉 冊
cot
k zh
= k s␦ .
2
共5兲
As explained in Ref. 5, Tc is the para-/ferroelectric transition temperature for the thin film. Putting ⳵ P / ⳵t = 0 in Eq.
共1兲, the corresponding stationary equation of the order parameter P can be written as
Lc P ⬅ − A共Tc − TC0兲P −
1
d2 P
P+D 2 ,
dz
␧d
共6兲
where Lc is the parabolic operator evaluated at the bifurcation point T = Tc, in the neighborhood of which the solution P
is small, if assumed continuous. Following Nicholis and
Prigogine,16 both P and ␥ ⬅ T − Tc can be expanded in a
power series in terms of a small perturbation ␭ from the
critical point:
P = ␭P1 + ␭2 P2 + ¯ ,
␥ = T − T c = ␭ ␥ 1 + ␭ 2␥ 2 + ¯ .
共7兲
This expansion is more flexible than the seemingly more
natural one in which P is expanded in a power series of 共T
− Tc兲.16 More importantly, it allows fractional power dependence of P on 共T − Tc兲. By substituting the expansion 共7兲 into
Eq. 共6兲, and equating coefficients of equal powers of ␭, a set
of relations of the following form can be obtained:
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044114-3
J. Appl. Phys. 100, 044114 共2006兲
B. Wang and C. H. Woo
共8兲
k = 1,2, ¯ ,
L c P k = a k,
which have to be satisfied together with the boundary and the
initial conditions:
Pk
dPk
= ⫿ ,
dz
␦
h
z= ± .
2
共9兲
The first several coefficients ak are
FIG. 1. Schematic of bifurcation diagram for ferroelectric thin film.
a1 = 0,
a2 = A␥1 P1 ,
共10兲
It is easy to check that under the boundary condition 共9兲, the
operator Lc is self-adjoint, and
共Lc P*, Pk兲 =
冕
h/2
PkLc P*dz
−h/2
= 共P*,Lc Pk兲
冕
h/2
共11兲
P*akdz = 0,
−h/2
where P* is the solution of the following homogeneous equation:
Lc P* = 0,
P*
dP*
= ⫿ ,
dz
␦
h
z= ± ,
2
共12兲
Eq. 共11兲 can be used to determine the coefficients ␥i. Then
from the second relation of Eq. 共7兲, one can determine ␭ as a
function of 共T − Tc兲. Substituting the resulting ␭ into the first
relation of Eq. 共7兲 and solving the inhomogeneous equations
共9兲 results in an explicit expression for the solution P.
Let us consider the ␦ ⬎ 0 case, which is satisfied by most
ferroelectrics. Equation 共4兲 dictates that the transition temperature of the thin film is lower than that of the bulk material. The solution of Eq. 共12兲 can be written as
P* = M cos共␤z兲 + N sin共␤z兲,
共13兲
where
␤ = 冑A共TC0 − Tc兲/D − 1/␧dD.
共14兲
Suppose the P is distributed symmetrically along the thickness direction, i.e., dP* / dz = 0 at z = 0, then N = 0, and the
solution 共13兲 becomes
P* = M cos共␤z兲.
共15兲
Similarly, the solution of the first equation in Eq. 共8兲 can be
obtained in the same form as P*. Thus,
P1 = M cos共␤z兲.
共16兲
By using the orthogonal condition in Eq. 共11兲, ␥1 and ␥2 can
be determined,
␥1 = 0
since
冕
h/2
−h/2
P1 P*dz =
冕
h/2
−h/2
P21dz ⫽ 0.
共17兲
冕
冕
P41dz
−h/2
h/2
.
共18兲
P21dz
−h/2
If the corresponding bulk ferroelectrics transition is of
second order, i.e., B ⬎ 0, B + 4GQ2 is also positive, and substitution of Eq. 共18兲 into 共7兲 gives
␭= ±
= 共P*,ak兲
=
B + 4GQ
A
␥2 = −
a3 = A␥1 P2 + A␥2 P1 + 共B + 4GQ2兲P31 .
2
h/2
冉 冊
Tc − T
− ␥2
1/2
for T ⬍ Tc ,
共19兲
i.e., the solution is defined in the supercritical 共ferroelectric兲
regime. According to Theorem 3.1 for the bifurcation of a
simple eigenvalue,17 the bifurcating solutions are both asymptotically stable in the supercritical regime. The phase
diagram for this case is shown in Fig. 1共a兲. Thus, if the bulk
ferroelectrics exhibits second-order phase transition, their
thin-film counterparts will also exhibit the same order of
phase transition, and the universal critical exponents remain
unchanged.
On the other hand, if the corresponding transition in the
bulk material is first order, i.e., B ⬍ 0, then depending on the
magnitude of the elastic self-energy of the transformation
volume 4GQ2, which is positive definite and has a magnitude
of the same order as B, B + 4GQ2 can be either positive or
negative. When 4GQ2 is sufficiently large, B + 4GQ2 is positive definite, as in BaTiO3 and PbTiO3, then a first-order
bulk ferroelectrics becomes second-order one in a rigidly
constrained thin film. This is consistent with the results of
Pertsev et al.18 For the two classical perovskite ferroelectrics,
BaTiO3 and PbTiO3, these authors also found that the twodimensional clamping of the film could result in a change of
transition order. Nevertheless, the results of Ref. 18 should
be interpreted noting the assumption that the gradient terms
in the free energy were negligible.
If the transformation volume satisfies B + 4GQ2 = 0, the
solution will include higher order terms of expansions, and
thus the critical exponents will differ from their bulk counterpart. If B + 4GQ2 is negative, substitution of Eq. 共18兲 into
Eq. 共7兲 gives
␭⯝ ±
冉 冊
T − Tc
␥2
1/2
for T ⬎ Tc ,
共20兲
i.e., the solution is defined only in the subcritical 共paraelectric兲 regime, and both branches are unstable. Physically,
there is a maximum value of Tmax above which only one
solution is admissible. Therefore, the subcritical branches
have to turn, at some value Tmax, in the direction of decreas-
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044114-4
J. Appl. Phys. 100, 044114 共2006兲
B. Wang and C. H. Woo
ing temperature 关Fig. 1共b兲兴. Thus, the solution of P near the
critical point can be obtained in the form as
P = P 1␭
=
冦
±M cos共␤z兲
±M cos共␤z兲
冉 冊
冉 冊
Tc − T
− ␥2
1/2
T − Tc
␥2
1/2
, B + 4GQ2 ⬎ 0 and T ⬍ Tc
, B + 4GQ2 ⬍ 0 and T ⬎ Tc .
冧
共21兲
It is clear from Eq. 共21兲 that for rigidly constrained thin-film
ferroelectrics the order of transitions is independent of its
thickness, but depends on the elastic self-energy of the transformation strain under the constraint of the substrate. Furthermore, we note that the second solution in Eq. 共21兲 is
unstable, and the actual transition is discontinuous. The behavior of the self-polarization P in the foregoing discussions
agrees completely with that derived earlier using linear stability analysis.5 Of course, the solution of P in Eq. 共21兲 cannot be obtained in Ref. 5.
冦
关A + 3共B + 4GQ2兲M 2 cos2共␤z兲/␥2兴共T − Tc兲
␹s−1 = for T ⬍ Tc ,
A共T − Tc兲 for T ⬎ Tc .
冧
共26兲
The average value of ␹s cannot be integrated analytically,
such as the average value of the reciprocal. Thus,
具␹s−1典 =
1
h
冕
h/2
␹s−1dz
−h/2
冦
3␪共Tc − T兲
=A 1−
h
冋
冕
冕
h/2
关
cos2共␤z兲dz兴2
−h/2
h/2
−h/2
cos 共␤z兲dz
4
冧
共T − Tc兲
= A 1 − 3␪共Tc − T兲
⫻
册
4 sin2共␤h兲 + 8␤h sin共␤h兲 + 4␤2h2
共T − Tc兲,
␤h sin共2␤h兲 + 8␤h sin共␤h兲 + 6␤2h2
共27兲
IV. DIELECTRIC SUSCEPTIBILITY OF
FERROELECTRIC FILMS NEAR THE CRITICAL POINT
The susceptibility ␹total of a ferroelectric material measures the response of the total 共i.e., observable兲 polarization
in an external field, and is defined by
␹total =
冏 冏
⳵ Ptotal
⳵Eext
共22兲
.
Eext→0
For the present boundary conditions, it can be shown from
the laws of electrostatics,
␹total =
1
␬d
冉冏 冏
⳵P
⳵Eext
Eext→0
冊
+ ␹d ⬅
1
共␹ + ␹d兲,
␬d s
共23兲
where ␬d is the dielectric constant 共=␧d / ␧0兲.
It is sufficient to calculate ␹s, which is much larger than
␹d near the transition temperature. ␹s can be calculated via
the total free energy as usual,
␹s−1 =
⳵2 f
⳵ P2
= A共T − TC0兲 +
1
+ 3共B + 4GQ2兲P2 + 5CP4 + D␤2
␧d
= A共T − Tc兲 + 3共B + 4GQ2兲P2 + 5CP4 .
where ␪共x兲 = 0 for x ⬍ 0, and ␪共x兲 = 1 for x ⬎ 0 is the step
function. Equation 共27兲 may be rewritten in the form
具␹s−1典−1 =
A−1⌰−1共h兲
.
共T − Tc兲
共28兲
Equation 共28兲 shows that, on both sides of the transition
temperature, the Curie-Weiss law holds, and that the average
susceptibility of the film diverges with a critical exponent of
−1. In addition, the corresponding Curie parameter
−1 −1
␬−1
d A ⌰ 共h兲 also changes sign discontinuously across the
transition temperature. Both aspects of its behavior are the
same as the bulk counterpart. However, for films of finite
thickness, the magnitude of the Curie parameter varies according to the thickness, leading to asymmetric divergences
across the transition temperature. Indeed, the Curie parameter is independent of the film thickness 共⌰−1 = 1兲 on the
paraelectric side 共T ⬎ Tc兲, a well known fact, but depends on
the film thickness on the ferroelectric side. Taking the limits
in Eq. 共27兲 and using the relations between ␤ and h derived
in Ref. 5, it can be shown that the value of ⌰−1 depends on
the film thickness h, approaching −关1 − 共8␦ / 3h兲兴 for thick
films 共h Ⰷ ␦兲, and −0.5关1 + 共h2 / 5␦2兲兴 for thin films 共h Ⰶ ␦兲.
共24兲
In deriving Eq. 共24兲, we have used Eqs. 共4兲 and 共13b兲 and the
following relations:
冉 冊 冋 冉 冊册
冉 冊
⳵2 ⳵ P
⳵ P2 ⳵z
2
=
⳵ ⳵z ⳵ ⳵ P
⳵ P ⳵ P ⳵z ⳵z
=2
2
⳵ ⳵2 P
⳵z ⳵3 P
=
2
⳵ P ⳵z2
⳵ P ⳵z3
共25兲
For positive B + 4GQ2, substitution of the first equation
of 共21兲, and ␤ in Eq. 共13b兲, into Eq. 共24兲 yields the inverse
susceptibility near Tc,
FIG. 2. Normalized susceptibility vs the temperature.
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044114-5
J. Appl. Phys. 100, 044114 共2006兲
B. Wang and C. H. Woo
TABLE I. Phenomenological parameters of PbTiO3 共in cgs unit兲.
␦
Material
To0
共K兲
A
10−5
D
10−15
nm
␣
PbTiO3
763
8.37
3
5
210
transition temperature was derived using a perturbation approach. It was found that the order of the transition and the
critical exponents can be changed by the constraint of the
substrate. A Curie-Weiss-type relation of dielectric permittivity was obtained in both the supercritical and subcritical regions. It was found that in paraelectric state, the Curie parameter is independent of the film thickness, and remains the
same as their bulk counterpart. In the ferroelectric state, on
the other hand, the magnitude of the Curie parameter of the
film decreases rapidly as its thickness decreases.
FIG. 3. The normalized Curie-Weiss parameter −关⌰共h兲兴−1 vs the film thickness h for PbTiO3.
Accordingly, for a film thickness of h = 8␦, the Curie param−1
eter ␬−1
of the ferroelectric phase would have dropped
d ⌰
33%. These characteristics are shown in Fig. 2, where the
normalized susceptibility, i.e., ⌰−1 / 共T − Tc兲 as a function of
temperature, is compared between thick films 共h → ⬁兲 and
thin films 共h → 0兲. We note that 具␹total典 also has a similar
relation to T − Tc, as discussed in the foregoing.
From the free energy expression in Ref. 5, the contribution from the upper and lower surfaces of the ferroelectric
thin film increases the susceptibility according to
⌬␹ =
2␦
,
D
共29兲
which is negligible near the transition temperature.
In the following, we consider the example of a well
known ferroelectric PbTiO3, the material constants of which
are shown in Table I.5 The normalized Curie parameter
−关⌰共h兲兴−1 is plotted against the film thickness h in Fig. 3, in
which the Curie temperature was obtained as a function of h
by using Eqs. 共4兲 and 共5兲. It can be seen that the magnitude
of the normalized Curie parameter does decrease rapidly
from 1 to 0.5 with decreasing thickness. Indeed, for films of
PbTiO3 about 40 nm 共=8␦兲, the ferroelectric transition is accompanied with a decrease of the Curie parameter of about
33%. Similar behavior is also found experimentally for BST
films.19 A review on the properties of ferroelectric materials
was given in references.20–22
V. SUMMARY AND CONCLUSIONS
In this paper, the explicit solution of the steady-state
Ginzburg-Landau equation for a ferroelectric film near the
ACKNOWLEDGMENTS
This project was supported by grants from the Research
Grants Council of the Hong Kong Special Administrative
Region 共PolyU5322/04E and 5312/03E兲 Natural Science
Foundation 共10572兲.
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